Algebraic K-theory and Motivic Cohomology

韦伯建议在代数K理论和动机理论的相关领域进行研究。首先，他提出要完成一个定理的证明，这个定理描述了K-理论和动机上同调与代数几何和数论的相关领域的关系。 该项目的第二部分是将新的motivic技术应用于代数几何中处理奇点的长期问题。本课题的第三部分将研究动机的内部结构，试图解释Chow群的行为，Chow群是代数簇的经典不变量。 他将尝试建立Rost簇的存在，完成Bloch-Kato猜想的验证。这个猜想将K-理论与数论和Etale上同调联系起来。 他还将研究之间的关系的奇点的品种，其K-理论和cdh-上同调，使用最近开发的上同调技术。这涉及到几个经典问题的K-理论环的整数在全球领域一直是一个焦点的研究在K-理论在过去的35年，特别是在过去的十年。此外，他将研究动机的结构，特别是切片过滤的结构，作为攻击关于Chow群和更高Chow群消失的假设的一种方式。

Weibel proposes to do research in the related fields of algebraic K-theory and motivic theory. First, he proposes to complete the proof of a theorem which describes the relationship that K-theory and motivic cohomology have with the related fields of algebraic geometry and number theory. A second part of the project is to apply new motivic techniques to long-standing problems in algebraic geometry dealing with singularities. The third part of the project will be to study the internal structure of motives in an attempt to explain the behavior of Chow groups, which are classical invariants of algebraic varieties.Specifically, Weibel proposes three strands of research. He will try to establish the existence of Rost varieties, completing the verification of the Bloch-Kato Conjecture. This conjecture links K-theory to number theory and etale cohomology. He will also study the relationship between the singularities of a variety, its K-theory and its cdh-cohomology, using recently developed cohomological techniques. This involves several classical questions about the K-theory of rings of integers in global fields which have been a focal point of research in K-theory for the last 35 years, especially in the last decade. In addition, he will study the structure of Motives, especially the structure of slice filtration, as a way of attacking conjectures about vanishing of Chow groups and higher Chow groups.